Energy levels of Bloch electrons inmagnetic fields

Douglas R. Hofstadter

Douglas Hofstadter

“Goedel, Escher, Bach: An Eternal Golden Braid”

Magnetic fields

No magnetic field:

H =p2

2m+ V (r)

With magnetic field:

H =1

2m

(p− q

cA)2

+ V (r)

Magnetic fields

No magnetic field:

H =p2

2m+ V (r)

With magnetic field:

H =1

2m

(p− q

cA)2

+ V (r)

Magnetic fields and Bloch electrons

Peierls substitution:

~k→ ~k− q

cA

Lattice

Landau Gauge

B = Bz B = ∇×A

A = Bxy

Landau Gauge

B = Bz B = ∇×A

A = Bxy

Landau Gauge

B = Bz B = ∇×A

A = Bxy

Tight-Binding model (no field)

Each lattice site carries a localised orbital, denotedby |n,m〉.The Hamiltonian induces transitions betweenneighbouring lattice points.

H |n,m〉 = E0 |n,m〉+

t(|n+ 1,m〉+ |n− 1,m〉

+ |n,m+ 1〉+ |n,m− 1〉)

Tight-Binding model (no field)

Each lattice site carries a localised orbital, denotedby |n,m〉.The Hamiltonian induces transitions betweenneighbouring lattice points.

H |n,m〉 = E0 |n,m〉+

t(|n+ 1,m〉+ |n− 1,m〉

+ |n,m+ 1〉+ |n,m− 1〉)

Effect of magnetic field

Transition matrix element gains a path dependence:

t→ t · exp(−i q

~cA · l

)

Effect of magnetic field

Using A = Bxy:

Transition amplitude along x unaffected:

t→ t

Transition amplitude along y gains x-dependence:

t→ t · exp(−i q

~cBx · (±a)

)

Effect of magnetic field

Using A = Bxy:

Transition amplitude along x unaffected:

t→ t

Transition amplitude along y gains x-dependence:

t→ t · exp(−i q

~cBx · (±a)

)

Dimensionless parameters. . .

α = a2B · qhc

Then, along y:

t→ t · exp (−i · 2παn)

Dimensionless parameters. . .

α = a2B · qhc

Then, along y:

t→ t · exp (−i · 2παn)

Tight-Binding Hamiltonian (with field)

H |n,m〉 = E0 |n,m〉+ t(|n+ 1,m〉+ |n− 1,m〉)

+ te−i2παn |n,m+ 1〉+ tei2παn |n,m− 1〉

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